The spate of arrests, details of under-the-table bribes to teenagers and the expected downfall of one of the sport’s best-known coaches has triggered uncomfortable soul-searching among the institutions at the heart of college basketball, including internal reviews by more than two dozen schools of their own prominent programs.

The spate of arrests, details of under-the-table bribes to teenagers and the expected downfall of one of the sport’s best-known coaches has triggered uncomfortable soul-searching among the institutions at the heart of college basketball, including internal reviews by more than two dozen schools of their own prominent programs.